Generalization of torsion points on Jacobian of genus 2 over finite fields (with respect to the theta divisor)

Question

Let $J(C)$ be the jacobian of a hyperelliptic curve $C$ of genus 2 defined over finite field $\mathbb{F}_q$. Let $\Theta$ be the image of the curve on the Jacobian under the embedding $P \mapsto P - \mathcal{O}$, which is also known as the theta divisor.

Do we know something about the structure of the following set: $$ J(C)_{\Theta}[n] := \{D \in J(C) : nD \in \Theta \}$$ Does this set have a name? Like "Theta n-torsion points"?

Clearly, the n-torsion points defined as $$ J(C)[n] := \{D \in J(C) : nD = \mathcal O \}$$ is a subset of $J(C)_{\Theta}[n]$.

I could not find many papers explicitly studying this set, the structure and cardinality. I found one paper Division polynomials and multiplication formulae of Jacobian varieties of dimension 2 by N. Kanayama where he defines this set in Section 3.2.2.

Is it correct/wrong to call it generalization of torsion points?

I am interested in the cardinality of the set $$ J(C)_{\Theta}[n] \cap J(C)_{\Theta}[m], m, n \in \mathbb{Z}, m \neq n$$.

Answer 1

The set $J(C)_{\Theta}[n]$ has the structure of a smooth irreducible algebraic curve, and the restriction of $J(C)\xrightarrow{\times n } J(C)$ to $C$ defines a morphism $J(C)_{\Theta}[n]\rightarrow C$ which is a finite etale cover with Galois group $J(C)[n]$. I don't think it is good to think of $J(C)_{\Theta}[n]$ as generalized torsion points. I think it's better to think of $J(C)_{\Theta}[n]$ as a curve which is an unramified cover of $C$. For example, while $J(C)[n]$ has $n^4$ points over $\bar{\mathbb{F}}_q$ (when $q$ is coprime to $n$), $J(C)_{\Theta}[n]$ has infinitely many $\bar{\mathbb{F}}_q$-points. Also by Riemann-Hurwitz the genus of $J(C)_{\Theta}[n]$ is $n^4+1$, so the Hasse-Weil bound tells you something about its $\mathbb{F}_q$-points.

For your second question about $J(C)_{\Theta}[m] \cap J(C)_{\Theta}[n]$, I will interpret this as a scheme theoretic intersection. (In other words, incorporating multiplicities.) We may calculate this using some facts about intersection theory on surfaces and line bundles on abelian varieties. As noted before, the curve $J(C)_{\Theta}[m]$ is the pullback of the divisor $C\subset J(C)$ (embedded using your implicitly chosen point $\mathcal{O}$). By the theorem of the square, we have the following linear equivalence of divisors on $J(C)$: $$J(C)_{\Theta}[n] \sim \frac{n^2+n}{2} C + \frac{n^2-n}{2} [-1]^*C.$$ Here $[-1]^*C$ is the image of $C$ under the $[-1]$ map. Since $[-1]^*C$ is algebraically equivalent to $C$ (even linearly equivalent if you choose $\mathcal{O}$ to be a Weierstrass point), we conclude that $$ (J_{\Theta}(C)[m],J_{\Theta}(C)[n]) = m^2n^2(C,C). $$ By the adjunction formula (and the fact that the canonical bundle of $J(C)$ is trivial): $(C,C) = 2p_a(C)-2 = 2$.

Conclusion: When counted with multiplicity, $J_{\Theta}(C)[m]$ and $J_{\Theta}(C)[n]$ intersect in $2m^2n^2$ points.